11 THE SPACES L1 AND L2 167 10. Suppose that the measure ju. is bounded and that /x(X) = 1. Let u be a £i-measur- able mapping of X into itself which leaves invariant the measure p, (Section 13.9, Problem 24). Then for every function/e J5f1(X, /x) the class U -f=f° u depends only on the class of/, hence defines an endomorphism of L1, also denoted by /t> U /, such that NI(£/ /) = N!(/). The restriction of U to L2 is a unitary operator on L2. (a) If P is the orthogonal projection of L2 on the subspace of vectors which are invariant under £/, denned in Problem 9, show that N^/)2 <; (/| P /) for all/e L2. (b) Show that for each/e L1 and each e > 0 there exists an integer n such that, for all integers m > 0, (1 m+n-l \ p./_i E £/*/<*. rt * = m / (c) Deduce that, for each measurable subset A of X and each e > 0, there exists an integer n > 0 with the property that for each integer m > 0, there exists an integer k such that m^k <zm-{- n 1 and p,(A n u~k(AJ) ^ (p,(A))2 e ("Khintchine's statistical recurrence theorem "). (d) For each/e 3?^ and each integer n, put Show that the limit I m-l m-*oo m, fcasO (where P -/denotes a function in the class P /) exists almost everywhere and that lim Fw dp, = 0. (Use Birkhoffs ergodic theorem.) (e) Let/e^(X, fM) be such that/(x)^ 0 almost everywhere. Show that, for almost all x e X, either f(x) = 0 or (P /)(*) > 0. (Consider the set N of points x e X at which (P /)(*) is either undefined or equal to zero.) 1. Let X be a metrizable compact space and let u: X-*X be a homeomorphism. The set I of measures ^0 on X of total mass 1 and invariant under u is then a nonempty vaguely compact subset of M(X) (Section 13.4, Problem 8). A point x e X is said to be quasi-regular (relative to u) if the sequence of measures converges vaguely as n -> + °o (necessarily to a measure />tx e I). Let Q denote the set of quasi-regular points of X. A point x e Q is said to be ergodic (relative to u) if the measure p,x is ergodic (Section 13.9, Problem 13). Let E be the set of ergodic points x e Q. A point x e Q is said to be dense ifx belongs to the support of p,x. Let D be the set of dense points xeQ, The points belonging to R Ğ E n D are said to be regular (relative to p). (a) Show that the complement of R (and hence also the complement of Q, E and D) is negligible with respect to any invariant measure vel. (To show that Q has measure 1 for any measure v e I, apply BirkhofFs ergodic theorem to the functions belonging to a dense sequence in ^(X), To show that D has measure 1, consider a denumerable basis (UĞ) for the topology of X, and for each pair (m, n) such that Om c Un, a continuousfore the series with general term aHfn(x) converges absolutely